MathDB

3

Part of 2020 Bundeswettbewerb Mathematik

Problems(2)

Sum of product of distances is constant

Source: Bundeswettbewerb Mathematik 2020, Round 1 - Problem 3

11/17/2020
Let ABAB be the diameter of a circle kk and let EE be a point in the interior of kk. The line AEAE intersects kk a second time in C≠AC \ne A and the line BEBE intersects kk a second time in D≠BD \ne B.
Show that the value of AC⋅AE+BD⋅BEAC \cdot AE+BD\cdot BE is independent of the choice of EE.
geometrygeometry proposeddistancesdiameter
Points moving on two lines with constant speed, common point of circles

Source: Bundeswettbewerb Mathematik 2020, Round 2 - Problem 3

11/17/2020
Two lines mm and nn intersect in a unique point PP. A point MM moves along mm with constant speed, while another point NN moves along nn with the same speed. They both pass through the point PP, but not at the same time.
Show that there is a fixed point Q≠PQ \ne P such that the points P,Q,MP,Q,M and NN lie on a common circle all the time.
geometrygeometry proposedcircumcirclecyclic quadrilateralmoving points